Energy decay and periodic solution for the wave equation in an exterior domain with half-linear and nonlinear boundary dissipations

We first consider the wave equation in an exterior domain Ω in R^N with two separated boundary parts Γ₀, Γ₁. On Γ₀, the Dirichlet condition u |_Γ0 = 0 is imposed, while on Γ₁, Neumann type nonlinear boundary dissipation ∂ u / ∂ ν = - g (u_t) is assumed. Further, a 'half-linear' localized dissipation is attached on Ω. For such a situation we derive a precise rate of decay of the energy E (t) for solutions of the initial boundary value problem. We impose no geometrical condition on the shape of the boundary ∂ Ω = Γ₀ ∪ Γ₁. Secondly, when a T periodic forcing term works we prove the existence of a T periodic solution on R under an additional growth assumption on ρ (x, v) and g (v).